How do I model this curvy spiral staircase?
Well I think this is just supposed to be a spiral staircase, a fairly simple helix.

One way to model it would be to make a square "U" shape with three edges (e.g. delete one edge of a plane).  Rotate it so it's as shown in the image below (make sure to apply your rotation) and you'll want to offset the origin, in Edit Mode, just select the vertices and move G them to one side, which will leave the origin in place.

Then just add a Screw modifier as shown. I have set the Angle to negative to get the helix to twist the direction I want.



Edit: as per @Duarte Farrajota Ramos, an Edge Split modifier would be a good addition to distinguish between the sharp angles and the smoother surfaces.